In this question, we consider some variations of the production externality example discussed in Lecture 21. To begin, suppose Farm 1 has cost function C₁ (q₁ , z ) = 7 4log(1 + z ) + (1/2)q₁² + w_D z , where q₁ is Farm 1's quantity of soybeans produced, z is the quantity of herbicide used, and w_D is the cost of the herbicide. Suppose the price of Farm 1's soybeans is p1 .

(a) Prove that Farm 1 cost function is a convex function of q1 and z . [Hint: Since ²C₁/q₁ z = 0, it is enough to verify that C1 is a convex function of q1 and of z separately.]

(b) What are the profit maximizing choices of q1 and z if p1 = 4 and w_d = 1? What are the profit maximizing choices of q1 and z if p1 = 4 and w_d = 5?

Farm 2 has cost function C₂ (q₂ , z ) = (z/15)q² + (1/2) q₂² , where q2 is Farm 2's quantity of soybeans produced and z is the quantity of herbicide used by Farm 1. Suppose the price of Farm 2's soybeans is p2 .

(c) Suppose p1 = 4, p2 = 5.2, and w_D = 1. What is Farm 2's profit maximizing choice of q2 ? How much would Farm 2 be willing to pay Farm 1 in exchange for Farm 1 choosing z = 0?

(d) This will not be enough to induce Farm 1 to choose z = 0. What is the minimum payment necessary to induce Farm 1 to choose z = 0?

(e) Explain why there is a tax rate that the government can impose on the use of herbicides that increases total profits above the no tax level. Do not attempt to calculate it! Suppose, unexpectedly, the government were to return the tax revenue raised to Farm 1 (the point of this is to ensure that Farm 1's choice of z is determined by the tax rate). Does the final outcome Pareto dominate the no tax allocation?